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<!DOCTYPE html>
<html data-require="math math-format word-problems">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Comparing improper fractions and mixed numbers</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div id="same-whole">
<div class="vars" data-ensure="M_DENOM !== I_DENOM">
<var id="WHOLE">randRange( 1, 5 )</var>
<var id="WHOLE_2">WHOLE</var>
<var id="M_NUM">randRange( 1, 9 )</var>
<var id="M_DENOM">randRange( M_NUM + 1, 10 )</var>
<var id="M_DENOM_REDUCED">M_DENOM/ getGCD( M_NUM, M_DENOM )</var>
<var id="M_NUM_REDUCED">M_NUM/ getGCD( M_NUM, M_DENOM )</var>
<var id="M_NUM_2">randRange( 1, 9 )</var>
<var id="M_DENOM_2">randRange( M_NUM_2 + 1, 10 )</var>
<var id="M_REDUCED_NUM">M_NUM_2 / getGCD( M_NUM_2, M_DENOM_2 )</var>
<var id="M_REDUCED_DENOM">M_DENOM_2 / getGCD( M_NUM_2, M_DENOM_2 )</var>
<var id="I_NUM">WHOLE_2 * M_REDUCED_DENOM + M_REDUCED_NUM</var>
<var id="I_DENOM">M_REDUCED_DENOM</var>
<var id="LCM">getLCM( M_DENOM_REDUCED, I_DENOM )</var>
<var id="F1">LCM / M_DENOM_REDUCED</var>
<var id="F2">LCM / I_DENOM</var>
<var id="BECOMES_1">F1 === 1 ? "remains as" : "becomes"</var>
<var id="BECOMES_2">F2 === 1 ? "remains as" : "becomes"</var>
<var id="M_AS_I">M_DENOM_REDUCED*WHOLE+M_NUM_REDUCED</var>
<var id="SOLUTION">(function() {
if ( (M_AS_I*F1) > (I_NUM*F2) ) {
return "&gt;";
} else if ( (M_AS_I*F1) === (I_NUM*F2) ) {
return "=";
}
else {
return "&lt;";
}
})()
</var>
</div>
<div class="problem">
<p>Fill in the blank.</p>
<p>
<code><var>WHOLE</var>\ <var>fraction( M_NUM, M_DENOM, false, true )</var></code>
____<code><var>fraction( I_NUM, I_DENOM, false, true )</var></code>
</p>
</div>
<p class="solution"><code><var>SOLUTION</var></code></p>
<ul class="choices" data-category="true">
<li><code>&lt;</code></li>
<li><code>=</code></li>
<li><code>&gt;</code></li>
</ul>
<div class="hints">
<p>First, let's convert the mixed number to an improper fraction with the same denominator.</p>
<p>To get the numerator of the improper fraction, multiply the denominator (<strong><var>M_DENOM_REDUCED</var></strong>) by the whole number (<strong><var>WHOLE</var></strong>) and add the numerator (<strong><var>M_NUM_REDUCED</var></strong>).</p>
<p class="hint_purple"><strong><code><var>M_DENOM_REDUCED</var> \cdot <var>WHOLE</var>+<var>M_NUM_REDUCED</var> = <var>M_AS_I</var></code></strong></p>
<p>We can write the mixed number as an improper fraction with numerator <strong><var>M_AS_I</var></strong> and denominator <strong><var>M_DENOM_REDUCED</var></strong>.</p>
<p>Now we need to compare <code><var>fraction ( M_AS_I, M_DENOM_REDUCED, false, true )</var></code> to <code><var>fraction ( I_NUM, I_DENOM, false, true )</var></code>.</p>
<div data-if="M_DENOM_REDUCED !== I_DENOM" data-unwrap>
<p>It is easier to compare these two fractions when they have the same denominator.</p>
<p>Their smallest common denominator is the LCM of <var>M_DENOM_REDUCED</var> and <var>I_DENOM</var>.</p>
<p><code>\lcm(<var>M_DENOM_REDUCED</var>, <var>I_DENOM</var>) = <var>LCM</var></code></p>
<div>
<p>The first fraction <var>BECOMES_1</var> <code>\dfrac{<var>M_AS_I * F1</var>}{<var>LCM</var>}</code>.</p>
<p>The second fraction <var>BECOMES_2</var> <code>\dfrac{<var>I_NUM * F2</var>}{<var>LCM</var>}</code>.</p>
</div>
</div>
<p>We see that <code>\dfrac{<var>M_AS_I * F1</var>}{<var>LCM</var>} <var>SOLUTION</var> \dfrac{<var>I_NUM * F2</var>}{<var>LCM</var>}</code>.</p>
</div>
</div>
<div id="different-whole">
<div class="vars" data-ensure="M_DENOM !== I_DENOM">
<var id="WHOLE">randRange( 1, 5 )</var>
<var id="WHOLE_2">randRange( 1, 5 )</var>
<var id="M_NUM">randRange( 1, 9 )</var>
<var id="M_DENOM">randRange( M_NUM + 1, 10 )</var>
<var id="M_DENOM_REDUCED">M_DENOM/ getGCD( M_NUM, M_DENOM )</var>
<var id="M_NUM_REDUCED">M_NUM/ getGCD( M_NUM, M_DENOM )</var>
<var id="M_NUM_2">randRange( 1, 9 )</var>
<var id="M_DENOM_2">randRange( M_NUM_2 + 1, 10 )</var>
<var id="M_REDUCED_NUM">M_NUM_2 / getGCD( M_NUM_2, M_DENOM_2 )</var>
<var id="M_REDUCED_DENOM">M_DENOM_2 / getGCD( M_NUM_2, M_DENOM_2 )</var>
<var id="I_NUM">WHOLE_2 * M_REDUCED_DENOM + M_REDUCED_NUM</var>
<var id="I_DENOM">M_REDUCED_DENOM</var>
<var id="LCM">getLCM( M_DENOM_REDUCED, I_DENOM )</var>
<var id="F1">LCM / M_DENOM_REDUCED</var>
<var id="F2">LCM / I_DENOM</var>
<var id="BECOMES_1">F1 === 1 ? "remains as" : "becomes"</var>
<var id="BECOMES_2">F2 === 1 ? "remains as" : "becomes"</var>
<var id="M_AS_I">M_DENOM_REDUCED*WHOLE+M_NUM_REDUCED</var>
<var id="SOLUTION">(function() {
if ( (M_AS_I*F1) > (I_NUM*F2) ) {
return "&gt;";
} else if ( (M_AS_I*F1) === (I_NUM*F2) ) {
return "=";
}
else {
return "&lt;";
}
})()
</var>
</div>
<div class="problem">
<p>Fill in the blank.</p>
<p>
<code><var>WHOLE</var>\ <var>fraction( M_NUM, M_DENOM, false, true )</var></code>
____<code><var>fraction( I_NUM, I_DENOM, false, true )</var></code>
</p>
</div>
<p class="solution"><code><var>SOLUTION</var></code></p>
<ul class="choices" data-category="true">
<li><code>&lt;</code></li>
<li><code>=</code></li>
<li><code>&gt;</code></li>
</ul>
<div class="hints">
<p>First, let's convert the mixed number to an improper fraction with the same denominator.</p>
<p>To get the numerator of the improper fraction, multiply the denominator (<strong><var>M_DENOM_REDUCED</var></strong>) by the whole number (<strong><var>WHOLE</var></strong>) and add the numerator (<strong><var>M_NUM_REDUCED</var></strong>).</p>
<p class="hint_purple"><strong><code><var>M_DENOM_REDUCED</var>\cdot<var>WHOLE</var>+<var>M_NUM_REDUCED</var> = <var>M_AS_I</var></code></strong></p>
<p>We can write the mixed number as an improper fraction with numerator <strong><var>M_AS_I</var></strong> and denominator <strong><var>M_DENOM_REDUCED</var></strong>.</p>
<p>Now we need to compare <code><var>fraction ( M_AS_I, M_DENOM_REDUCED, false, true )</var></code> to <code><var>fraction ( I_NUM, I_DENOM, false, true )</var></code>.</p>
<div data-if="M_DENOM_REDUCED !== I_DENOM" data-unwrap>
<p>It is easier to compare these two fractions when they have the same denominator.</p>
<p>Their smallest common denominator is the LCM of <var>M_DENOM_REDUCED</var> and <var>I_DENOM</var>.</p>
<p><code>\lcm(<var>M_DENOM_REDUCED</var>, <var>I_DENOM</var>) = <var>LCM</var></code></p>
<div>
<p>The first fraction <var>BECOMES_1</var> <code>\dfrac{<var>M_AS_I * F1</var>}{<var>LCM</var>}</code>.</p>
<p>The second fraction <var>BECOMES_2</var> <code>\dfrac{<var>I_NUM * F2</var>}{<var>LCM</var>}</code>.</p>
</div>
</div>
<p>We see that <code>\dfrac{<var>M_AS_I * F1</var>}{<var>LCM</var>} <var>SOLUTION</var> \dfrac{<var>I_NUM * F2</var>}{<var>LCM</var>}</code>.</p>
</div>
</div>
</div>
</div>
</body>
</html>
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